Natural Numbers
All the positive integers from 1, 2, 3,……, ∞.
Whole Numbers
All the natural numbers including zero are called Whole Numbers.
Integers
All negative and positive numbers including zero are called Integers.
Rational Numbers
A number is called Rational if it can be expressed in the form p/q where p and q are integers
(q> 0). It includes all natural, whole number and integers.Example: 1/2, 4/3, 5/7,1 etc.
1.2 PROPERTIES OF RATIONAL NUMBERS:
- Closure Property
This shows that the operation of any two same types of numbers is also the same type or not.
a). Whole Numbers
If p and q are two whole numbers then
| Operation | Addition | Subtraction | Multiplication | Division |
|
Whole number |
p + q will also be the whole number. |
p – q will not always be a whole number. |
pq will also be the whole number. |
p ÷ q will not always be a whole number. |
|
Example |
6 + 0 = 6 |
8 – 10 = – 2 |
3 × 5 = 15 |
3 ÷ 5 = 3/5 |
|
Closed or Not |
Closed |
Not closed |
Closed |
Not closed |
b). Integers
If p and q are two integers then
|
Operation |
Addition |
Subtraction |
Multiplication |
Division |
|
Integers |
p+q will also be an integer. |
p-q will also be an integer. |
pq will also be an integer. |
p ÷ q will not always be an integer. |
| Example | – 3 + 2 = – 1 | 5 – 7 = – 2 | – 5 × 8 = – 40 | – 5 ÷ 7 = – 5/7 |
| Closed or not | Closed | Closed | Closed | Not closed |
c). Rational Numbers
If p and q are two rational numbers then
|
Operation |
Addition |
Subtraction |
Multiplication |
Division |
| Rational Numbers | p + q will also be a rational number. | p – q will also be a rational number. | pq will also be a rational number. | p ÷ q will not always be a rational number |
| Example | p ÷ 0
= not defined |
|||
|
Closed or not |
Closed |
Closed |
Closed |
Not closed |
- Commutative Property
This shows that the position of numbers does not matter i.e. if you swap the positions of the numbers then also the result will be the same.
a). Whole Numbers
If p and q are two whole numbers then
| Operation | Addition | Subtraction | Multiplication | Division |
| Whole number | p + q = q + p | p – q ≠ q – p | p × q = q × p | p ÷ q ≠ q ÷ p |
| Example | 3 + 2 = 2 + 3 | 8 –10 ≠ 10 – 8 – 2 ≠ 2 | 3 × 5 = 5 × 3 | 3 ÷ 5 ≠ 5 ÷ 3 |
| Commutative | yes | No | yes | No |
b). Integers
If p and q are two integers then
| Operation | Addition | Subtraction | Multiplication | Division |
| Whole number | p + q = q + p | p – q ≠ q – p | p × q = q × p | p ÷ q ≠ q ÷ p |
| Example | True | 5 – 7 = – 7 – (5) | – 5 × 8 = 8 × (–5) | – 5 ÷ 7 ≠ 7 ÷ (-5) |
| Commutative | yes | No | yes | No |
c). Rational Numbers
If p and q are two rational numbers then
| Operation | Addition | Subtraction | Multiplication | Division |
| Example |
= + (
|
( ) –
|
=
|
|
| Example | True | 5 – 7 = – 7 – (5) | – 5 × 8 = 8 × (–5) | – 5 ÷ 7 ≠ 7 ÷ (-5) |
| Commutative | yes | No | yes | No |
- Associative Property
This shows that the grouping of numbers does not matter i.e. we can use operations on any two numbers first and the result will be the same.
a). Whole Numbers
If p, q and r are three whole numbers then
| Operation | Addition | Subtraction | Multiplication | Division |
| Whole number | p + (q + r) = (p + q) + r | p – (q – r) = (p – q) – r | p × (q × r) = (p × q) × r | p ÷ (q ÷ r) ≠ (p ÷ q) ÷ r |
| Example | 3 + (2 + 5) = (3 + 2) + 5 | 8 – (10 – 2) ≠ (8 -10) – 2 | 3 × (5 × 2) = (3 × 5) × 2 | 10 ÷ (5 ÷ 1) ≠ (10 ÷ 5) ÷ 1 |
| Associative | yes | No | yes | No |
- Integers
If p, q and r are three integers then
| Operation | Integers | Example | Associative |
| Addition | p + (q + r) = (p + q) + r | (– 6) + [(– 4)+(–5)] = [(– 6) +(– 4)] + (–5) | Yes |
| Subtraction | p – (q – r) = (p – q) – r | 5 – (7 – 3) ≠ (5 – 7) – 3 | No |
| Multiplication | p × (q × r) = (p × q) × r | (– 4) × [(– 8) ×(–5)] = [(– 4) × (– 8)] × (–5) | Yes |
| Division | p ÷ (q ÷ r) ≠ (p ÷ q) ÷ r
|
[(–10) ÷ 2] ÷ (–5) ≠ (–10) ÷ [2 ÷ (– 5)] | No |
- Rational Numbers
If p, q and r are three rational numbers then
| Operation | Integers | Example | Associative |
| Addition | p + (q + r) = (p + q) + r | =
= =
|
Yes |
| Subtraction | p – (q – r) = (p – q) – r | No | |
| Multiplication | p × (q × r) = (p × q) × r | = ×
Hence LHS = RHS |
Yes |
| Division | p ÷ (q ÷ r) ≠ (p ÷ q) ÷ r |
|
No |
The Role of Zero in Numbers (Additive Identity)
Zero is the additive identity for whole numbers, integers and rational numbers.
| Identity | Example | ||
| Whole number | a + 0 = 0 + a = a | Addition of zero to whole number | 2 + 0 = 0 + 2 = 2 |
| Integer | b + 0 = 0 + b = b | Addition of zero to an integer | False |
| Rational number | c + 0 = 0 + c = c | Addition of zero to a rational number | 2/5 + 0 = 0 + 2/5 = 2/5 |
The Role of one in Numbers (Multiplicative Identity)
One is the multiplicative identity for whole numbers, integers and rational numbers.
| Identity | Example | ||||
| Whole number | a ×1 = a | Multiplication of one to the whole number | 5 × 1 = 5 | ||
| Integer | b × 1= b | Multiplication of one to an integer | – 5 × 1 = – 5 | ||
| Rational Number | c × 1= c | Multiplication of one to a rational number |
|
||
Negative of a Number (Additive Inverse)
| Identity | Example | ||
| Whole number | a +(- a) = 0 | Where a is a whole number | 5 + (-5) = 0 |
| Integer | b +(- b) = 0 | Where b is an integer | True |
| Rational number | c + (-c) = 0 | Where c is a rational number |
|
Reciprocal (Multiplicative Inverse)
The multiplicative inverse of any rational number
Example :The reciprocal of is.
Distributivity of Multiplication over Addition and Subtraction for Rational Numbers
This shows that for all rational numbers p, q and r
- p(q + r) = pq + pr2. p(q – r) = pq – pr
Example:Check the distributive property of the three rational numbers 4/7,-( 2)/3 and 1/2.
Solution:Let’s find the value of
